Question: Determine how many solutions exist for the system of equations. ${2x+y = -5}$ ${2x+y = -10}$
Explanation: Convert both equations to slope-intercept form: ${2x+y = -5}$ $2x{-2x} + y = -5{-2x}$ $y = -5-2x$ ${y = -2x-5}$ ${2x+y = -10}$ $2x{-2x} + y = -10{-2x}$ $y = -10-2x$ ${y = -2x-10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -2x-5}$ ${y = -2x-10}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.